As far as I know, you can configure any controller to work with Dolphin. As it is, I've configured both a PS4 controller (via XInput and DS4Windows) and a Gamecube controller (with no software, only just the Dolphin config) for use with Dolphin, so I'm sure the Steam controller would work too.

Was gonna call my friends and get 'em all together
I was gonna go and start a band
But everything I wanna do, it's already been done
Why would I wanna go and do it again?